Choosing the right ratio · 选择合适的比值
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| sine rule/saɪn ruːl/ | 正弦定理 | zhèng xián dìng lǐ |
| cosine rule/ˈkəʊsaɪn ruːl/ | 余弦定理 | yú xián dìng lǐ |
| unit circle/ˈjuːnɪt ˈsɜːkl/ | 单位圆 | dān wèi yuán |
| periodic/ˌpɪərɪˈɒdɪk/ | 周期的 | zhōu qī de |
| radians/ˈreɪdɪənz/ | 弧度 | hú dù |
| bearing/ˈbeərɪŋ/ | 方位角 | fāng wèi jiǎo |
| angle of elevation/ˈæŋɡl ɒv ˌelɪˈveɪʃn/ | 仰角 | yǎng jiǎo |
| angle of depression/ˈæŋɡl ɒv dɪˈpreʃn/ | 俯角 | fǔ jiǎo |
Match an angle to its sides
- Right-triangle ratios relate a chosen acute angle to opposite, adjacent and hypotenuse lengths. Opposite and adjacent change when the chosen angle changes.
- Sine is opposite over hypotenuse, cosine adjacent over hypotenuse, and tangent opposite over adjacent. Identify the known and requested lengths before choosing a ratio.
Select the equation before calculating
- Draw and label the right triangle. The longest side opposite the right angle is the hypotenuse.
- With opposite and adjacent, use $\tan\theta=o/a$. Rearrange for the unknown; use inverse tangent if the angle is unknown, and check the angle mode.
Watch the ratios as the angle changes · 观察角度变化时各比值的变化
See why tangent grows without limit while sine and cosine never leave [−1, 1]. · 理解为何正切无界增长，而正弦和余弦始终在 [−1, 1] 内。
You know the adjacent side and want the opposite side. Which ratio do you use? · 已知邻边，要求对边，应使用哪个比值？
Tangent is opposite over adjacent — the only ratio that involves neither hypotenuse. · 正切是对边比邻边——唯一不涉及斜边的比值。
Extend the method beyond right triangles
- The sine rule 正弦定理 pairs sides with opposite angles: $a/\sin A=b/\sin B$. The cosine rule 余弦定理 uses $c^2=a^2+b^2-2ab\cos C$, where C is the included angle.
- The unit circle 单位圆 extends sine and cosine beyond acute angles. They are periodic 周期的, repeating every 360 degrees or $2\pi$ radians; tangent has a different period and undefined values.
From 40 m away, the angle of elevation of a tower top is 32°. Find the height in metres, to 1 decimal place. · 从距离 40 米处观测，塔顶的仰角为 32°。求塔高（单位：米），精确到 1 位小数。
h = 40 tan 32° = 25.0 m. Make sure the calculator is in degree mode. · h = 40 tan 32° = 25.0 m。确保计算器处于角度模式。
Put a trigonometry solution in the order that earns the method marks. · 按能获得方法分的顺序书写三角函数解题过程。
The ratio line is the marked step. A number with no ratio line above it earns nothing if it is wrong. · 比值等式是关键得分步骤。若无此步，后续数值即使正确也无分。
Keep the reference direction explicit
- A bearing 方位角 is clockwise from north and uses three digits in degrees. An angle of elevation 仰角 is upward from the horizontal, and an angle of depression 俯角 downward.
- Radians 弧度 measure angles with $180^\circ=\pi$ radians. Convert units or set the calculator correctly before substituting.
You know two sides of a triangle and the angle between them. Which rule finds the third side? · 已知三角形两边及其夹角，用哪条法则求第三边？
Two sides and the included angle is exactly what a² = b² + c² − 2bc cos A needs. Pythagoras needs a right angle. · 两边及夹角正是 a² = b² + c² − 2bc cos A 所需条件。勾股定理需要直角。
Two observations share one height. A tower seen at 35 degrees, then 50 degrees after moving 10 m closer, gives $h=d\tan35^\circ=(d-10)\tan50^\circ$. The resulting initial distance is about 24.245 m. Add an instrument height only when converting h to tower height above ground.
Write a bearing of seventy-five degrees in the correct three-figure form. · 将“七十五度”方位角写成正确的三位数形式。
Bearings always take three figures, so a leading zero is required. · 方位角总是三位数，因此需补前导零。
How many degrees is π radians? · π 弧度等于多少度？
Half a full turn. A full turn is 2π radians, or 360°. · 半圈。一整圈是 2π 弧度，或 360°。
Explain which data select the rule. Two sides and their included angle fit the cosine rule. A known side-angle opposite pair supports the sine rule. Sheet 1.5 combines triangle lengths, bearings and two observations.
With side-side-angle data, inverse sine may produce an acute angle while its supplement gives another valid triangle. Check the remaining angle sum before discarding that possibility. Do not assume a right angle that the question does not state.